On my forum (http://baseofarts.forumotion.com), the chatbox is visible, but you cant see any posts or names or anything. It just says Chatbox, Refresh, Archive, and Login at the top. You can type stuff, but nothing shows up.

Right, it's being a problem with mine too. Did you check your chat box's css? or the template? I can't really help you much from here apart from telling you to check on your codes and check your javascripts since I've got little experience with punBB

Seems to me that you have added a javascript code disabling your chatbox or when you added the theme somewhere in the CSS coding it confused your chatbox so what i consider is going through everything which will take a lot of time and find the error or just getting a new chatbox like XAT chat there are many more out there and disabling that one until someone comes along and helps you, ill try to find an answer to your problems

Well thats for you to decide. Whatever edits you have done in the templates is messing up the chatbox in my opinion. Save your edited templates in a word file. Remove the ones related to that index page & then check.